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COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK

  • KARL LORENSEN (a1)

Abstract

Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp. We show that if G is nilpotent, then the pro-p completion map induces an isomorphism for any discrete -module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map is an isomorphism for any discrete -module M of finite p-power order. Moreover, if G lacks any Cp-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

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References

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[1]Cutolo, G. and Smith, H., ‘A note on polycyclic residually finite-p groups’, Glasg. Math. J. 52(1) (2010), 137143.
[2]Kropholler, P., Private communication.
[3]Lennox, J. and Robinson, D., The Theory of Infinite Soluble Groups (Clarendon Press, Oxford, 2004).
[4]Linnell, P. and Schick, T., ‘Finite group extensions and the Atiyah conjecture’, J. Amer. Math. Soc. 20 (2007), 10031051.
[5]Robinson, D., ‘On the cohomology of soluble groups of finite rank’, J. Pure Appl. Algebra 6 (1975), 155164.
[6]Serre, J.-P., Galois Cohomology (Springer, Berlin, 1997).
[7]S̆mel’kin, A., ‘Polycyclic groups’, Sibirsk Mat. Zh. 9 (1968), 234235.
[8]Weigel, T., ‘On profinite groups with finite abelianizations’, Selecta Math. (N.S.) 13 (2007), 175181.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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